View source for List of transitive finite linear groups

In [[Mathematics|mathematics]], especially in areas of [[Abstract algebra|abstract algebra]] and [[Finite geometry|finite geometry]], the '''list of transitive finite linear groups''' is an important classification of certain highly symmetric [[Group action (mathematics)|actions]] of [[Finite group|finite group]]s on [[Vector space|vector space]]s.
The [[Solvable group|solvable]] finite [[2-transitive group|2-transitive group]]s were classified by [[Biography:Bertram Huppert|Bertram Huppert]].<ref>{{Citation | last1=Huppert | first1=Bertram |  title=Zweifach transitive, auflösbare Permutationsgruppen | doi=10.1007/BF01160336 |mr=0094386 | year=1957 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=68 | pages=126–150}}</ref> The [[Classification of finite simple groups|classification of finite simple groups]] made possible the complete classification of finite doubly transitive permutation groups. This is a result by Christoph Hering.<ref>{{Citation | last1=Hering | first1=Christoph | title=Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II | doi=10.1016/0021-8693(85)90179-6 |mr=780488 | year=1985 | journal=Journal of Algebra | issn=0021-8693 | volume=93 | issue=1 | pages=151–164| doi-access=free }}</ref>  A finite 2-transitive group has a [[Socle (mathematics)|socle]] that is either a vector space over a [[Finite field|finite field]] or a non-abelian primitive [[Simple group|simple group]]; groups of the latter kind are almost simple groups and described elsewhere<!-- haha, where? someday at [[Almost simple group|almost simple group]]? -->.  This article provides a complete <!-- and irredundant? --> list of the finite 2-transitive groups whose socle is [[Elementary abelian group|elementary abelian]].

Let <math>p</math> be a prime, and <math>G</math> a subgroup of the [[General linear group|general linear group]] <math>GL(d,p)</math> acting transitively on the nonzero vectors of the ''d''-dimensional vector space <math>(F_p)^d</math> over the [[Finite field|finite field]]<math>F_p</math> with ''p'' elements.

== Infinite classes ==

There are four infinite classes of finite transitive linear groups.

* <math>G \leq \Gamma{}L(1,p^d);</math>
* <math>G \triangleright SL(a,q)\text{ and }p^d=q^a;</math>
* <math>G \triangleright Sp(2a,q)\text{ and }p^d=q^{2a};</math>
* <math>G \triangleright G_2(q)',\  p^d=q^6\text{ and }p=2.</math>

Notice that the exceptional [[Group of Lie type|group of Lie type]] ''G''<sub>2</sub>(''q'') is usually constructed as the automorphism groups of the split [[Octonion|octonion]]s. Hence, it has a natural [[Representation (mathematics)|representation]] as a subgroup of the 7-dimensional [[Orthogonal group|orthogonal group]] O(7,&nbsp;''q''). If ''q'' is even, then the underlying [[Quadratic form|quadratic form]] polarizes to a degenerate symplectic form. Factoring out with the radical, one obtains an [[Isomorphism|isomorphism]] between O(7,&nbsp;''q'') and the [[Symplectic group|symplectic group]] Sp(6,&nbsp;''q''). The subgroup of Sp(6,&nbsp;''q'') which corresponds to ''G''<sub>2</sub>(''q'')′ is transitive.

In fact, for ''q''>2, the group ''G''<sub>2</sub>(''q'') = ''G''<sub>2</sub>(''q'')′ is simple. If ''q''=2 then ''G''<sub>2</sub>(2)′ ≅ PSU(3,3) is simple with index 2 in ''G''<sub>2</sub>(2).

== Sporadic finite transitive linear groups ==

These groups are usually classified by some typical [[Normal subgroup|normal subgroup]], this normal subgroup is denoted by ''G''<sub>0</sub> and are written in the third column of the table. The notation 2<sup>1+4</sup><sub>−</sub> stands for the extraspecial group of minus type of order 32 (i.e. the extraspecial group of order 32 with an odd number (namely one) of quaternion factor).

All but one of the sporadic transitive linear groups <math>G</math> yield a primitive permutation group <math>p^d:G</math> of degree at most 2499. In the computer algebra programs GAP and MAGMA, these groups can be accessed with the command <code>PrimitiveGroup(p^d,k);</code> where the number ''k'' is the ''primitive identification'' of <math>p^d:G</math>. This number is given in the last column of the following table.

Seven of these groups are sharply transitive; these groups were found by [[Biography:Hans Zassenhaus|Hans Zassenhaus]] and are also known as the multiplicative groups of the Zassenhaus [[Near-field (mathematics)|near-field]]s. These groups are marked by a star in the table.

{| class="wikitable" border="1"
|-
! Condition on <math>p</math>
! Condition on <math>d</math>
! <math>G_0</math>
! Primitive identification of <math>p^d:G</math>
|-
| <math>p=5</math>
| <math>d=2</math>
| <math>SL(2,3)</math>
| 15*, 18, 19
|-
| <math>p=7</math>
| <math>d=2</math>
| <math>SL(2,3)</math>
| 25*, 29
|-
| <math>p=11</math>
| <math>d=2</math>
| <math>SL(2,3)</math>
| 39*, 42
|-
| <math>p=23</math>
| <math>d=2</math>
| <math>SL(2,3)</math>
| 59*
|-
| <math>p=11</math>
| <math>d=2</math>
| <math>SL(2,5)</math>
| 56*, 57
|-
| <math>p=19</math>
| <math>d=2</math>
| <math>SL(2,5)</math>
| 86
|-
| <math>p=29</math>
| <math>d=2</math>
| <math>SL(2,5)</math>
| 106*, 110
|-
| <math>p=59</math>
| <math>d=2</math>
| <math>SL(2,5)</math>
| 84*
|-
| <math>p=2</math>
| <math>d=4</math>
| <math>A_6</math>
| 16, 17
|-
| <math>p=2</math>
| <math>d=4</math>
| <math>A_7</math>
| 20
|-
| <math>p=3</math>
| <math>d=4</math>
| <math>SL(2,5)</math>
| 124, 126, 127, 128
|-
| <math>p=3</math>
| <math>d=4</math>
| <math>2^{1+4}_-</math>
| 71, 90, 99, 129, 130
|-
| <math>p=2</math>
| <math>d=6</math>
| <math>PSU(3,3)</math>
| 62, 63
|-
| <math>p=3</math>
| <math>d=6</math>
| <math>SL(2,13)</math>
| 396
|-
|}

This list is not explicitly contained in Hering's paper. Many books<ref>{{Citation | last1=Huppert | first1=Bertram | last2=Blackburn | first2=Norman | title=Finite groups. III. | publisher=Springer-Verlag | location=Berlin-New York | series=Grundlehren der Mathematischen Wissenschaften | isbn=3-540-10633-2 |mr=0650245 | year=1982 | volume=243}}</ref><ref>{{Citation | last1=Johnson | first1=Norman L. | last2=Jha | first2=Vikram | last3=Biliotti | first3=Mauro | title=Handbook of finite translation planes | publisher=Chapman & Hall/CRC | location=Boca Raton | series=Pure and Applied Mathematics | isbn=978-1-58488-605-1 |mr=2290291 | year=2007 | volume=289}}</ref> and papers give a list of these groups, some of them an incomplete one. For example, Cameron's book<ref>{{Citation | last1=Cameron | first1=Peter J. | title=Permutation Groups | publisher={{wipe|Cambridge University Press}} | series=London Mathematical Society Student Texts | isbn=978-0-521-65378-7 | year=1999 | volume=45 | url-access=registration | url=https://archive.org/details/permutationgroup0000came }}</ref> misses the groups in line 11 of the table, that is, containing <math>SL(2,5)</math> as a normal subgroup.

== References ==
{{reflist}}

[[Category:Permutation groups]]

{{Sourceattribution|List of transitive finite linear groups}}